MAC-CPTM Situations Project
Situation 50: Connecting Factoring with the Quadratic
Formula
Prepared at University of Georgia
Center for Proficiency in Teaching Mathematics
13 September 2006 – Erik Tillema
Edited at Pennsylvania State University
Mid-Atlantic Center for Mathematics Teaching and Learning
21 February 2007 – Heather Godine
Prompt
Mr. Jones suspected his students saw no direct connection between the work they
had done on factoring quadratics and the quadratic formula.
Commentary
Any quadratic polynomial can be represented as a product of two binomials.
Quadratic polynomials of the form x 2  bx  c can be represented as a product of
the form x  mx  n , where m,n may have real or complex values. This
situation deals specifically with those quadratic polynomials that can be
represented as a product of
the form x  mx  n  where m,n   . Partitioned
2
area models emphasize
 the relationship between the sum x  bx  c and the

product x  mx  n  in cases for which x is a positive real number. In each of
the foci in this Situation, we
will assume x is positive.


Mathematical Foci

Mathematical Focus 1
Partitioned area models can be used to represent x 2  bx  c  x  mx  n
where m,n   .

Consider the quadratic polynomial x 2  3x  2 , and a partitioned area model (figure 1).


Figure 1
The process of factoring involves re-expressing a sum as a product. Hence, by
Situation <50> <FactQuadForm> <070221>
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composing a rectangle from the partitioned area model, the sum of the areas can be
expressed as a product of the lengths of the sides of the rectangle (figure 2).
Figure 2
Since the area of the rectangle can be represented as the product of the lengths of the
sides of the rectangle or as the sum of the partitioned areas, x 2  3x  2  x 1x  2.
Mathematical Focus 2
Partitioned area models can be used to represent x 2  a2  x  ax  a where

a  .

Consider the quadratic polynomial x 2  9 and a partitioned area model (figure 3).


Figure 3
To factor the quadratic polynomial x 2  9 , compose a rectangle from the partitioned area
model (figure 4).

Figure 4
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Since the area of the rectangle can be represented as the product of the lengths of the
sides of the rectangle or as the sum of the partitioned areas, x 2  9  x  3x  3.
Mathematical Focus 3
Partitioned area models can be used to represent the technique of completing
the square for quadratic polynomials of the form x2  bx .
Now consider quadratic polynomials of the form x 2  bx such that a square can be
composed from a partitioned area model representing x 2  bx  c .

6x and a partitioned area model (figure 5).
Consider the quadratic polynomial x 2  


Figure 5
To compose a square, construct two adjacent rectangles: One with length x and width
x+3 and another with length x and width 3 (figure 6).
Figure 6
Composing a square requires adding 9 additional units of area in the form of a 3x3
square (figure 7).
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Figure 7
Hence, x 2  6x  9  x  3x  3  x  3 .
2
Mathematical Focus 4
Partitioned area models can be used to represent x 2  bx  c  x  mx  n
where m,n   .

Consider the quadratic polynomial x 2  6x  7 and a partitioned area model (figure 8).


Figure 8
The partitioned area model is re-expressed in figure 9; however, a rectangle has not
been composed.
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Figure 9
Since x 2  6x  7  x 2  6x  9 2  x  3  2 , remove a square with two units of area
2
from the area model representing x  3 (figure 10).
2


Figure 10
To factor the quadratic polynomial x 2  6x  7 , compose a rectangle from the partitioned
area model (figure 11)

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Figure 11
Since the area of the rectangle can be represented as the product of the lengths of the
sides of the rectangle or as the sum of the partitioned areas,
x 2  6x  7  x  3  2 x  3  2 .



Mathematical Focus 5
Partitioned area models can be used to represent x 2  bx  c  x  mx  n
where m,n   .


Consider the quadratic polynomial x 2  bx  c and a partitioned area model (figure 12).


… (b x-units)
…. (c square units)
Figure 12
To factor the quadratic polynomial x 2  bx  c , compose a rectangle from the partitioned
area model (figures 13 and 14).

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Figure 13
Figure 14
Since the area of the rectangle can be represented as the product of the lengths of the
sides of the rectangle or as the sum of the partitioned areas,
2
2
x 2  bx  c  (x  b2  b4  c )(x  b2  b4  c ).




b  b 2  4ac
can be used to determine solutions to a quadratic
2a
equation of the form ax 2  bx  c  0. When a  1, the quadratic equation becomes
x 2  bx  c  0 , and applying the quadratic formula gives solutions
b  b 2  
4c b
b 2  4c b
b2
x




 c . Using the zero product property, solutions
2
2
4  2
4
2
2
to the equation x 2  bx  c  (x  b2  b4  c )(x  b2  b4  c )  0 can be determined:
The quadratic formula x 
x  b2 
x   b2 

b2
4
 c  0 and x  b2 
2
b
4
b2
4
 c  0 , resulting in x   b2 
b2
4
 c and
c .



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